Global linearizable actions on topological manifolds

TL;DR

This paper investigates the structure of affine homeomorphisms on topological aspherical manifolds, revealing their Lie group properties and foliation structures, especially for closed manifolds.

Contribution

It establishes that the connected component of affine homeomorphisms forms a solvable Lie group and describes its foliation structure, extending understanding of symmetries on topological manifolds.

Findings

$Aff(M)_0$ acts locally freely on $M$ for closed manifolds.

$Aff(M)_0$ is a solvable Lie group.

$Aff(M)_0$ is nilpotent if $M$ is a polynomial manifold.

Abstract

Let $M$ be a finite dimensional topological aspherical manifold whose universal cover is ${\bf R}^n$. In this paper, we study $Aff(M)$, the subgroup of the group of homeomorphisms of $M$, whose elements can be lifted to affine transformations of ${\bf R}^n$. We show that if $M$ is closed, the connected component $Aff(M)_0$ of $Aff(M)$ acts locally freely on $M$. We deduce that $Aff(M)_0$ is a solvable Lie group, and is nilpotent if $M$ is a polynomial manifold. We study the foliation defined by the orbits of $Aff(M)_0$ if $dim(Aff(M)_0)=dim(M)-1$.